linear fractional transformations

A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.

A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.

If is a linear transformation then . Say that . If then .

If and then the equation is not solvable and so has no fixed points in . It does however have as a fixed point on the Riemann sphere.

If and then is the identity.

If then is a fixed point in the otherfixed point is because if because otherwise is not a fixed point.

A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.

The question asks about linear transformations, but the thread is headed "linear fractional transformations", in other words those of the form . In that case, a fixed point must satisfy . This is clearly a quadratic equation for z, except when b=c=0 and a=d, when it becomes identically true.